Problem 5

Let $N$ be a positive integer.

1. Express $\frac{7x+4}{x^{3}+3x^{2}+2x}$ in the form $\frac{A}{x}+\frac{B}{x+1}+\frac{C}{x+2}$, where $A$, $B$, and $C$ are real constants to be determined.

2. Hence simplify

   $$\frac{7N+9}{N^{2}+3N+2}+\sum_{r=1}^{N}\frac{7r+4}{r^{3}+3r^{2}+2r}.$$